Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimddvcbv | Structured version Visualization version GIF version |
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40633. The equivalent of this theorem without the bound variable change is rexlimddv 3290. Usage of this theorem is discouraged because it depends on ax-13 2389, see rexlimddvcbvw 40634 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rexlimddvcbv.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) |
rexlimddvcbv.2 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) |
rexlimddvcbv.3 | ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexlimddvcbv | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimddvcbv.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | |
2 | rexlimddvcbv.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | |
3 | rexlimddvcbv.2 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | |
4 | 2, 3 | rexlimdvaacbv 40633 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓)) |
5 | 1, 4 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |